YoVDO

Enumerative Geometry and the Quantum Torus

Offered By: IMSA via YouTube

Tags

Tropical Geometry Courses Lie Algebras Courses Algebraic Geometry Courses Toric Varieties Courses Enumerative Geometry Courses

Course Description

Overview

Explore enumerative geometry and the quantum torus in this comprehensive lecture by Michael Polyak from the Israel Institute of Technology. Delve into tropical geometry as a piecewise linear approach to algebraic geometry, examining how tropical curves serve as planar metric graphs with specific balancing, slope rationality, and integrality requirements. Investigate a generalization of tropical curves that removes slope rationality and integrality constraints. Learn about classical enumerative problems in algebraic geometry, focusing on counting complex or real rational curves through point collections in toric varieties. Discover how this counting procedure relates to cycle construction on rigid pseudotropical curve moduli, and explore the connection between these cycles and Lie algebras. Examine the relationship between complex and real curve counting and the quantum torus Lie algebra. Delve into more complex counting invariants, such as Gromov-Witten descendants, and their connection to the super-Lie structure on the quantum torus. The lecture covers topics including an introduction to enumerative and tropical geometry, tropical curves as weighted balanced graphs, the correspondence theorem, refined tropical counting, pseudotropical curves, weights and enumeration, rigid marked curves, and evaluation maps and compactification.

Syllabus

Intro
Intro to enumerative and tropical geometry
Tropical curves as weighted balanced graphs
The correspondence theorem
A refined tropical count
Questions
Pseudotropical curves
Weights and enumeration
Rigid marked curves
Evaluation maps and a compactification


Taught by

IMSA

Related Courses

An Introduction to smooth Manifolds
Indian Institute of Science Bangalore via Swayam
Group Theory methods in Physics
Indian Institute of Technology Bombay via Swayam
Poisson Brackets, Non-Canonical Hamiltonian Systems and Euler's Rigid Body Equations
Ross Dynamics Lab via YouTube
The Ubiquity of ADE Graphs, and the Mutation and Numbers Games - Math Seminars
Insights into Mathematics via YouTube
The J Function, SL(2) and the Jacobi Identity - Universal Hyperbolic Geometry
Insights into Mathematics via YouTube